We give here an alternative formulation of the transformation
in terms of position eigenstates, following but improving on the
work of Ref.[1]}. The formulation allows the transformation
to be seen more directly as a symmetry transformation within the Hilbert space of
\({\mathcal {H}}_{\!\scriptscriptstyle A}\otimes {\mathcal {H}}_{\!\scriptscriptstyle B}\otimes {\mathcal {H}}_{\!\scriptscriptstyle C}\) ,
with the initial and final frames of reference taken as the `states'
\(\left|{\bf 0}\right\rangle _{\!\scriptscriptstyle A}\)  and \(\left|{\bf 0}\right\rangle _{\!\scriptscriptstyle B}\) ,
i.e. the zero vectors. We introduce the use of the zero vector
under the following considerations. A zero vector of course has no
observable physical properties. Any operator acts on it trivially. That
corresponds exactly to the idea that a frame of reference does not
see itself as a dynamical object, hence cannot have a state with any
nontrivial observable properties. We emphasize that it is no enough
that the description of a composite state of \(BC\)  as observed from
\(A\)  has no nontrivial content for its own position,
\(\hat{x}_{\!\scriptscriptstyle A}^{\scriptscriptstyle (A)}\) , when the quantum spatial translation
is concerned. For the case one may think the position eigenstate with
eigenvalue zero may do. First of all, the idea of writing that state of
\(BC\)  as observed from \(A\)  as a vector within
\({\mathcal {H}}_{\!\scriptscriptstyle A}\otimes {\mathcal {H}}_{\!\scriptscriptstyle B}\otimes {\mathcal {H}}_{\!\scriptscriptstyle C}\) 
should be a notion independent of which particular quantum
reference frame transformation one may want to formulate. At
least it is more desirable to have such a consistent formulation.
In the picture of Eq.(REF ) as presented in Ref.[2]}, there is no
\(\hat{p}_{\!\scriptscriptstyle A}^{\scriptscriptstyle (A)}\)  and \(\hat{p}_{\!\scriptscriptstyle B}^{\scriptscriptstyle (B)}\) 
to considered as there is no \(\hat{x}_{\!\scriptscriptstyle A}^{\scriptscriptstyle (A)}\)  and
\(\hat{x}_{\!\scriptscriptstyle B}^{\scriptscriptstyle (B)}\) . The zero position eigenstate has
no trivial \(\hat{p}_{\!\scriptscriptstyle A}^{\scriptscriptstyle (A)}\)  with quantum fluctuations,
and would lead to similar \(\hat{p}_{\!\scriptscriptstyle B}^{\scriptscriptstyle (B)}\)  after the
quantum translation. One can see that our formulation with the
zero vector is free from that and gives consistent results. Of course
whatever makes up the physical frame of reference would be
observed as a usual object from another frame of reference.
